Calculo 2 De dos variables_9na Edición - Ron Larson

6. fx, y x y 10x 12y 64
x
In Exercises 7–16, examine the function for relative extrema.
27. z e x sin y
7. fx, y 3x 2 2y 2 6x 4y 16
SECCIÓN 13.8 Extremos z de funciones de dos variables 961
8. f x, y 3x 2 2y 2 3x 4y 5
13.8 Extrema of Functions of Two Variables 961
13.8 Extrema of 8 Functions of Two Variables 961
9. fx, y x 2 5y 2 10x 10y 28
13.8 Extrema of Functions of Two Variables 961
35. Una función f tiene segundas derivadas parciales continuas en
6
10.
una región abierta que contiene el punto crítico (3, 7). La fun-
xtiene 2 1
Desarrollo de conceptos
35.
f
A
x,
function
y 2x 2 f has
2xy
continuous
y 2 2x
second
3
partial derivatives on an
11. 35. 4
zción open
function
region xy un f
containing
has mínimo 2 y 2 continuous 2x en the (3, ycritical 7) second y d point > partial 0 para 3,
derivatives
7 el . criterio The function
on de an WRITING ABOUT CONCEPTS
35. A function f has continuous second partial derivatives las an WRITING
segundas open derivadas parciales. Determinar el intervalo para
La figura muestra las 2 curvas de nivel de una función descono-
x,
12. zhas a minimum
region containing
5x 2 4xyat 3, y 2 7 ,
the
and
critical
16xd > 10 0 for
point
the Second
3, 7 . The
Partials
function 55. WRITING
ABOUT
The figure ABOUT
CONCEPTS
shows the CONCEPTS level curves for an unknown function
open region containing the critical point 3, 7 . The function Test. 55. The
f xy 3, 7 si f xx 3, 7 2 y f yy 3, 7 8.
fcida
figure
y f.
x, What,
shows
y. ¿Qué if any,
the
información, information
level curves
si can
for
es be
an
que given
unknown
hay about
function
has
Determine
a minimum
the
at alguna, puede
13. fx, y x 2 interval
3, 7 , and
y 2 for
d >
f
0 for
xy 3, 7
the
if
Second
f xx 3,
Partials
7 2
Test.
and
55. The figure shows the level curves for an unknown function f at the
has a minimum at 3, 7 , and d > 0 for the Second Partials Test. f
point
x, y
darse A?
. What,
acerca Explain
if any,
de f en your
information
el punto reasoning.
can be given about f at the
Determine
36. Una f
A? Explicar el razonamiento.
yy 3, función 7 8.
the interval for f
f tiene segundas derivadas xy 3, 7 if f
parciales xx 3, 7 2 and
f x, y . What, if any, information can be given about f at the
Determine the interval for f continuas en una
y
14. hx, f y x
región abierta 2 y
que 2 contiene 1 3 y 6 3π
yy 3, 7 8. 2 xy 3, 7 if f xx 3, 7 2 and point A? Explain your reasoning.
point A? Explain y your reasoning. y
36. A f yy function 3, 7 8. f has continuous el punto second crítico partial (a, derivatives b). Si f xx a, on bany
y x
y
15. 36. g fopen x,
yy a, function y b region tiene 4 f
containing signos has x continuous opuestos, y the critical
second ¿qué 16. implica point fx, partial y a, esto? b
derivatives x.
Explicar. If fy 2
xx a, b
on
and
an
y
y
36. A function f has continuous second partial derivatives on open
f
1
In En Exercises los ejercicios 17–20, 37 use a 42, a computer a) hallar algebra los puntos system críticos, to graph b) determinar
the
28. z
x 2
2 y 2 e 1 x2 y 2
CAS yy a,
region
b have
containing
opposite signs,
the critical
what is
point
implied?
a, b
Explain.
. If f xx a, b and
open region containing the critical point a, b . If f A
D
A
f yy a, b have opposite signs, what is implied? Explain. xx a, b and
x
f A
D A
surface yy los
a,
extremos
b have opposite
relativos,
signs,
c) indicar
what is
los
implied?
puntos
Explain.
and locate any relative extrema and saddle críticos points. en los
A
z
D A
CAS In Exercises 37– 42, (a) find the critical points, (b) test for
x
CAS cuales In
relative
Exercises el
extrema,
criterio 37– de las segundas derivadas parciales no concluyente,
y d) usar
4x (c)
42,
list
(a)
the
find
critical
the critical
points for
points,
which
(b)
the
test
Second
for
CAS In Exercises 37– 42, (a) find the critical points, (b) test for
17. relative
Partials z extrema, un sistema algebraico por computadora para
2
trazar la x
Test 2
función, y
fails,
(c) 2 1
and
list the
(d)
critical
use a computer
points for
algebra
which the
system
Second
B
relative extrema, (c) list the critical points for which the Second to
B C
Partials
graph the
Test
function,
fails, clasificando and
labeling
(d) use
any
cualesquiera a
extrema
computer
and
algebra puntos
saddle
extremos
points.
system toy
B
Partials Test fails, and (d) use a computer algebra system to
C
18. puntos fx, silla. y y 3 3yx 2 3y 2 3x 2 C
graph the function, labeling any extrema 1 and saddle points.
graph the function, labeling any extrema and saddle points.
19.
37. zf fx,
x, y
yx 2 x4y 3 2 e y 1 3 x2 y
37. fx, y x 3 y 2
Figure Figura for para 55 55 Figure Figura for para 56 56
20.
38. 37.
zf fx,
x, ey y xy x 3 y 3
3
6x 2 9y 2 12x
27y
19
Figure for 55 Figure for 56
La
Figure
figura
for
muestra
55
las curvas de
Figure
nivel de
for
una
56
38. fx, y x 3 y 3 6x 2 9y 2 12x 27y 19
56. The figure shows the level curves for an unknown función function des-
39. 38.
f fx,
x, y y x
x 3 1
1y 3 2 y
y 6x 2
4
4 2 9y 2 12x 27y 19
56. The
fconocida x,
figure
y . What, f(x, shows
if y). any,
the ¿Qué information
level información, curves
can be
an si given
unknown es que about hay function alguna, y
In 39. Exercises fx, y 21–28, x 1 examine 2 y 4 2
56. The figure shows the level curves an unknown 4function
f at the
the function for relative extrema
40. 39.
f fx,
x, y y x
x
x 1
1
1 2 2 y 4
y
y 2
2
2 2
f
points puede x, y .
A, darse What,
B, C, acerca if
and
any,
D? de information fExplain en los puntos your
can be
reasoning. A, given B, Cabout y D? Explicar f at the
f x, y . What,
and saddle points.
4
if any, information can be given about f at the
40. fx, y x 1 2 y 2 2
points el razonamiento. A, B, C, and D? Explain your reasoning.
41. 40.
f fx,
x, y y
x 23 23 x
y1 23 232
y 2 2
42.
f
fx,
x, y
y
x
x 2 y 2 223
points A,
21.
23
xB,
C, and D? Explain your reasoning.
41. hx, fx, y 80x x 23 80y y x 2 y 2 42. fx, y x 2 y In
En
Exercises
los ejercicios
57–59, a
sketch
59, dibujar
the
la
graph
gráfica
of
de
an
una
arbitrary
41. fx, y x 23 y 23
23
42. fx, y x 2 y 223
223
función
CAS In
In
22. En In Exercises los ejercicios 43 y hallar los puntos críticos de la función arbitraria
Exercises
f
29
que
and
satisfaga
30, examine
las condiciones
the function
dadas.
for
Decir
extrema
gx, y x
43 2 and
y 2 44,
x
find
y
the critical points of the function function
Exercises
f satisfying
57–59,
the
sketch
given
the
conditions.
graph
State
of an
whether
arbitrary
In Exercises 57–59, sketch the graph of an arbitrary si
the
la
In without
y, and,
Exercises
por from la forma the
43
form
and
de la of
44,
función, the
find
function,
the critical
determinar determine
points
si se whether
of the
presenta a
function function
relative
máximo
maximum
función
using f
tiene
has
satisfying
the any
extremos
derivative extrema
the given
o puntos
or tests, saddle
conditions.
and
silla.
points. use
(Hay
a
State
computer (There
whether
muchas
are
respuestas
correctas.)
algebra many
the
In Exercises 43 and 44, find the critical points of the function function f satisfying the given conditions. State whether the
23. and, gx, y xy
system from the
o un mínimo or a
form
relative
of the
relativo minimum
function,
en cada occurs
determine
punto. at each
whether
point.
a relative function
correct to answers.) graph
has any
the
extrema
surface.
or
(Hint:
saddle
By
points.
observation,
(There are
determine
many
and, from the form of the function, determine whether a relative function has any extrema or saddle points. (There are many
24. maximum hx, y or xa 2 relative 3xy minimum y 2 occurs at each point.
if
correct
it is possible
answers.)
maximum or a relative minimum occurs at each point.
correct answers.) for z to be negative. When is z equal to 0?)
43. f x, y, z x 2 y 3 2 z 1 2 57. f x f x x, y y > 0
and y f y fx, y x, y y < < 00para for all todo x, x, y . y.
25. 43.
fx, y, z x 2 x 2 xy y y 2 3 2 3x z y 1 2 57. f x x,
29. Todas xy >
las y primeras y segundas 30. derivadas x
z
2 parciales y
z
4 0 and f y x, y < 0 for all x, y . 2 2
44. 43. ffx, y, y, z de f son 0.
z
9x 2 x y 31 2 z z2 2 1 2 58. 57. All f x x, of y the > first 0 and and f y second x, y < partial 0 for all derivatives x, y . of f are 0.
44. fx, y, z 9 xy 1 z 2 2
58. All xof x f x 0, 2 the y
0 2 first and second partial derivatives f y 0, 0 0
2 yof 2 f are 0.
44. fx, y, z 9 xy 1 z 2 2
59. 58. f x
All of 0 the 0, first f y
and 0second 0 partial derivatives of f are 0.
En In los Exercises ejercicios 45–54, 4
a find 54, hallar the absolute los extremos extrema absolutos of the de function la función
In en la región R. (En cada caso, R contiene sus puntos fron-
59. f x 0, 0 0, f y 0, 0 0
Think About It
In 0, Exercises f
tera.) Utilizar un sistema algebraico por computadora y confirmar
los resultados.
x x, y , 31–34, f determine 0,
whether there is
over
Exercises
the region
45–54,
R. (In
find
each
the
case,
absolute
R contains
extrema
the boundaries.)
of the function
59. f x 0, 0
Use
< 0,
0, f y x
0,
<
0
0
0
a relative f x maximum, y a relative yx, minimum, y a saddle point, or
a computer algebra system to confirm your results.
> 0,
> , f y y > 0, y < 0
In Exercises 45–54, find the absolute extrema of the function
over the region R. (In each case, R contains the boundaries.) Use
< 0, x < 0
f < 0,
> insufficient x x, y
a computer algebra system to confirm your results.
information > 0, x > to 0 , determine f y x, y > 0, y < 0
over the region R. (In each case, R contains the boundaries.) Use
< 0, x < 0
f x x, y
the < 0, nature y > of 0 the function
a computer algebra system
y
to confirm your results.
f xx x, y f yy x, y <
f x, y
f 0, y f xy x, y para todo x, y.
45. fx, y x 2 4xy 5
at xx the y >
critical 0, f yy
x
point
> y 0 , <
x
0,
f y
0 , y
and x,
0 .
f y > 0, y < 0
xy x, < y0,
y 0 for > 0all x, y .
3
f
45. f x, y x 2 4xy 5
xx x, y > 0, f yy x, y < 0, and f xy x, y 0 for all x, y .
f
45. Rfx, y x, y x 2 :
31 4xyx 5
xx x, y > 0, f yy x, y < 0, and f xy x, y 0 for all x, y .
4, 0 y 2
31. f xx x 0 , y 0 9, f yy x 0 , y 0 4, f xy x 0 , y 0 6
R x, y : 1x
x 4, 0 y 2
46. fx, R y x, xy 2 : 1xy, xR 4, 0x, y y: x 2 2, y 1
32. CAPSTONE f xx x 0 , y 0 3, f yy x 0 , y 0 8, f xy x 0 , y 0 2
46. f x, y x 2 xy, R x, y : x 2, y 1
47. 46.
fx, y 12 x 2 3x xy, R2y
x, y : x 2, y 1
CAPSTONE
33. 60. CAPSTONE Para discusión
f xx
Consider x 0 , y 0
the 9, functions f yy x 0 , y 0 6, f xy x 0 , y 0 10
47. fx,
fR:
x,
y
The y
12
triangular
3x
2y
47. fx, y 12 3xregion 2y
60.
in the xy-plane with vertices 2, 0 , 34. Consider the functions
60. f
Considerar
xx
fx 0
x, , y 0
x25, the 2 las functions funciones yf 2
yy
yx 0
gx, , y 0
y 8, x 2 f xy
yx 2 0
., y 0 10
R:
R:
La
The
0, región 1
triangular
, andtriangular 1, 2
region
en
in
el
the
plano
xy-
plane
con
with
vértices
vertices
(2, 0),
2,
(0,
0
1)
,
R: The region in the xy-plane with vertices 2, 0 , f x, y x 2 y y gx, y x 2 y 2 .
(a) Show that both functions have a critical point at 0, 0 .
48. fx, y
0,
y1, 1
2
, and 1, 2
f x, y x 2 y 2
2 y g x, y x 2 y 2 .
0, 1 , 2x and 1, y 2
(a)
(b) Explain how f and g behave differently at this critical
48. fx,
fR:
x,
y
The y triangular 2x
2x
y
y 2
Show that both functions have a critical point at 0, 0 .
(a) Demostrar Show that que both ambas functions funciones have tienen a critical un punto point at crítico 0, 0 en .
48. fx, y y 2
region 2
in the xy-plane with vertices 2, 0 ,
(b) Explain
point.
how f and g behave differently at this critical
R:
R:
La
The
0, región 1
triangular
, andtriangular 1, 2
region
en
in
el
the
plano
xy-
plane
con
with
vértices
vertices
(2, 0),
2,
(0,
0
1)
,
(b)
(0,
Explain
0).
how f and g behave differently at this critical
R: The region in the xy-plane with vertices 2, 0 , b) Explicar point.
49. fx, y
0,
y1, 1
2
, and 1, 2
point. cómo f y g se comportan de manera diferente en
0, 1 , 3x and 2 1, 2y2
4y
este punto crítico.
49. fx,
fR:
x,
y
The y
3x
region 2 2
2y
in 2 the
2
4y
49. fx, y 3x 2 2y 2 xy-
4y plane bounded by the graphs of y x 2
R:
R:
La
The
andregión region
y 4en in
el
the
plano
xy-plane acotada
bounded
por
by
las
the
gráficas
graphs
de
of
y
y
x
x 2 2
y
True or False? In Exercises 61–64, determine whether the
R: The region in the xy-plane bounded by the graphs of y x ¿Verdadero o falso? En los ejercicios 61 a 64, determinar si la
50. y
and
True
y
4
4
2
statement
or False?
is true
In
or
Exercises
false. If it
61–64,
is false,
determine
explain why
whether
or give
the
an
fx, and y y2x 4
True or False? In Exercises 61–64, determine whether the
2xy y 2
statement
example
declaración
that
is true es
shows
verdadera or
it
false.
is false.
o If falsa. it is false, Si es explain falsa, explicar why or por give qué ano
50. fx,
fR:
x,
y
The y
2x
region
2xy
in the
y
xy-
2
statement is true or false. If it is false, explain why or give an
50. fx, y 2x 2xy y 2
plane 2
bounded by the graphs of y x 2 example dar un ejemplo that shows que it demuestre is false. que es falsa.
61. If f has a relative maximum at x then f
R:
R:
La
The
andregión region
en el plano acotada por las gráficas de y x 2 y
0 , y 0 , z 0 , x x 0 , y
y 1
in the xy-plane bounded by the graphs of y x 2 example that shows it is false.
0
R: The region in the xy-plane bounded by the graphs of y x 61. If
fSi tiene un máximo relativo en x , entonces ƒ x
(x , y 0
)
51. y
and
1
y x
f
0 ,
has
y
a relative
0 0.
maximum at x 0 , y 0 , z 0 , then f x x 0 , y
y 1
2
61. If f has a relative maximum at x 0
fx, and y x 2 2xy y 2 , R x, y : ƒ y
(x , y 0
) 0.
52. f x, y x, y x 2, ≤ 2, y 1
f y x 0 , y 0 0.
0 , y 0 , z 0 , then f x x 0 , y
y 1
0
51. fx, y x 2 2xy y 2 , R x, y : x 2, y ≤ 1 1 62. If f y fx 0 x, 0
y, 0 y 0
0. f y x 0 , y 0 0, then f has a relative maximum at
51.
fx, y x 2 2xy y 2 ,, R x, y :: x 2 x y 2 2, y8
1 62. If Si x
f
0 , x y
x(x 0 0 ,
, z y 0 0 .
) = f y (x x 0 , y 0 ) = 0, 0, entonces then f has f tiene a relative un máximo maximum relativo
52. fx,
f x,
y
y
x 2 2
2xy 4xy y 2 2 , R x,
x, y
y
: x 2 2
y 2 2 8
62. If f then f has a relative maximum 52.
≤ 8
en x 0 , (x y x x
0 , 0 , y zy 0 . 0 f y x 0 , y 0 0,
fx, y x 2 2xy y 2 , R x, y : x 2 y 2 8
, z 0
).
53. fx, y
63. Between x 0 , y 0 , z 0 any . two relative minima of f, there must be at least one
53. fx, f x, yy x 2 4xy
1) 4xyy 2 1
63. Between
relative Entre cualesquiera maximum
any two relative
of dos f. mínimos minima relativos of f, there de must f, aquí be at debe least estar oneal
53. fx, y x x 2 1)y y 2 1
63. Between any two relative minima of f, there must be at least one
1
relative
R x, y x: 2 0 xy 2 1, 10 y 1
64. If
menos
f is continuous
un maximum máximo of
for
relativo f.
relative maximum of all f. x and
de f.
y and has two relative minima,
R x, y y : 0 ≤ x ≤ 1, 1, 0 ≤ y ≤ 1 1
R x, y : 0 64.
4xyx 1, 0 y 1
If
then Si f
is es f
continuous
must continua have para at
for
least
all todo x
one
and x relative y y
and y tiene maximum.
has dos two mínimos relative minima,
64. If f is continuous for all x and y and has two relative relativos, minima,
54. fx, y
then entonces f must f debe have tener at least un one máximo relative relativo maximum. por lo menos.
54. fx, f x, yy x 2 4xy
14xyy 2 1
then f must have at least one relative maximum.
54. fx, y x x 2 11y y 2 11
R x, y x: 2 x 1 0, y 2 y 10, x 2 y 2 1
R x, y y : x ≥ 0, y ≥ 0, x 2 y 2 ≤ 1 1
R x, y : x 0, y 0, x 2 y 2 1
3